Is it permissible to use paint brushes made from mongoose hair? Is it permissible to use watercolours because they may contain ox gall but I'm not sure about it and if they contain ox gall then I also don't know how they're slaughtered.Can I use those products which may contain animal fat or similar things but I'm not sure whether it really

All perfect praise be to Allah, The Lord of the Worlds. I testify that there is none worthy of worship except Allah, and that Muhammad is His slave and Messenger.

The mongoose is one of the animals whose meat is unlawful for consumption. Ibn Qudaamah said in Al-Mughni: “The meat of the jackal, mongoose, and weasel is unlawful for consumption.” [End of quote]

Since the meat of the mongoose is unlawful for consumption, its dead body is considered ritually impure. It is also invalid to try to legalize its consumption by slaughtering it in accordance with the manner set by the Sharee‘ah. Jurists held different opinions regarding the ritual purity of the hair of a dead animal. Based on the opinion suggesting that it is ritually pure – which was the choice of Shaykhul-Islam Ibn Taymiyyah and the famous opinion of Imaam Maalik – there is nothing wrong with using a brush made of mongoose hair. However, there is no doubt that cautiousness and religious prudence lie in avoiding using it and rather using an alternative that is permissible by the agreement of scholars.

Shaykhul-Islam Ibn Taymiyyah said: “As for the bones, horns, and nails of a dead animal and whatever is of the same nature, such as its hooves and the like, as well as its hair, feathers, and fur, the scholars held three opinions regarding the ritual purity of these two categories. First: that they are all ritually impure, which is the famous opinion of Ash-Shaafi‘i as well as one of the opinions reported from Ahmad. Second: that the bones and their like are ritually impure, while the hair and its like are ritually pure, and this was the famous view of the Maaliki and Hanbali scholars. Third: that they are all ritually pure, and this was the opinion of Abu Haneefah and one of the opinions of the Maaliki and Hanbali scholars, and it is the correct opinion in this regard.” [End of quote]

As for the watercolors of which you suspect that they might contain the mentioned substance, the basic principle is that those colors are ritually pure because, as a basic rule, all things are ritually pure rather than impure. Therefore, there is nothing wrong with using them. If you know with certainty that they contain that substance, then if it is extracted from an ox that was slaughtered in accordance with the Sharee‘ah, there is no problematic aspect regarding the permissibility of using them in what is permissible, such as permissible drawings. If it is extracted from an ox that was not slaughtered according to the Sharee‘ah, then using those watercolors is subject to the ruling on using an object that has become ritually impure and ritually impure fluids. Some jurists held that it is permissible to use them in other than mosques, while others held that it is forbidden to use them at all.

Al-Mawsoo‘ah Al-Fiqhiyyah reads: “Using ritually impure fluids:

The Hanafi, Maaliki, and Hanbali scholars held that it is impermissible to use the grease or fat of a dead animal in caulking ships and the like, or lighting lamps or any other usage, except its skin if it is tanned ... According to the famous opinion of the Shaafi‘i school, the Shaafi‘i scholars held that it is permissible but disliked to use the ritually impure grease or fat of a dead animal or the (originally pure) oil and the like which is defiled with a ritual impurity that fell into it in (places) other than mosques … The author of Al-Majmoo‘ said: ‘It is permissible to caulk ships with the grease of a dead animal or feed it to dogs and birds, or feed the ritually impure food to animals.’

The Maaliki scholars, however, made a distinction between the intrinsically impure substances, like urine, and the originally pure substances that have become ritually impure. They held that it is permissible to use the originally pure substance that has become ritually impure in (places) other than mosques and bodies ... Al-Hattaab said in his commentary: ‘What he means with the defiled substance is what was originally pure, but was contaminated and became ritually impure…’” [End of quote; summarized]

So according to the opinions of the Shaafi’i and Maaliki scholars, it is permissible to use those colors that have been contaminated by the ox gall extracted from an ox that was not slaughtered according to the Sharee‘ah, and according to the opinions of the Hanbali and Hanafi scholars, it is impermissible.

With regard to your question, “Can I use those products which may contain animal fat or similar things but I am not sure whether it really does,” the answer is that it is permissible to use such products as long as you are not certain that they contain a ritually impure substance because the basic principle is that all things are ritually pure, not impure.

Allah Knows best.